Is Quarto solved? What a solver says about perfect play
Yes. Classic Quarto was solved by computer in 1998 by Luc Goossens. With perfect play from both players it's a draw, and nobody can force a win from the empty board. We wanted to know what happens when 2×2 squares also count as wins, as they do in Quads and in Quarto's advanced variant. So we solved 13,224 random endgames exactly under both rule sets. With seven pieces on the board, draws fell from 42.9% to 5.8% once squares counted.
About us: we make Quads, an independent game built on the same hand-your-opponent-the-piece mechanic. It isn't affiliated with Gigamic or the makers of Quarto. Everything below was computed with our own solver on our own game engine. The method and its limits are spelled out, and the raw numbers are published.
What "solved" means
A game is solved when a computer has worked out the result with perfect play from both sides. For Quarto the answer is a draw. Players often misread this in two ways:
- It doesn't make the game easy. A perfect draw means that at every turn, some choice avoids losing. It doesn't mean those choices are easy to find. There are 16 cells and up to 15 pieces to hand over on every turn, and the danger stays hidden until three pieces line up.
- It doesn't mean real games end in draws. One slip, like handing over a piece that finishes a line or leaving yourself no safe piece to give, usually decides the game. Most games between people are decisive.
Our question: what do 2×2 squares change?
Classic Quarto wins on 10 shapes: 4 rows, 4 columns and 2 diagonals. Many editions include an advanced variant where any 2×2 square of four matching pieces also wins, and Quads always plays that way. That adds 9 winning shapes, for 19 in total. Intuitively that should make draws rarer. We measured how much.
How we tested it
- Random endgames. We placed k random pieces on k random cells and threw out any board that had already been won. The player to move holds one random piece from the rest.
- Exact solving. An alpha-beta search with no depth limit and no node budget found each position's true result: a forced win for the player to move, a draw, or a forced loss. Each position was solved twice, once with classic lines only and once with lines plus squares.
- Checked. The solver agreed with an independent brute-force search on all 775 test positions, under both rule sets. Every run uses a fixed seed (20260915), so the numbers reproduce exactly.
An important limit: random positions aren't the positions strong players reach. Careful players avoid a lot of what random placement creates. Read these numbers as a comparison of the two rule sets on identical positions, not as the odds of winning your next game. We also didn't re-solve the full game from an empty board with squares.
Finding 1: squares turn draws into decisive games
Here's the result for the player to move with perfect play by both sides, on the same positions under each rule set:
| Pieces placed | Positions | Classic: win | Classic: draw | With squares: win | With squares: draw |
|---|---|---|---|---|---|
| 4 | 24* | 0% | 100% | 45.8% | 54.2% |
| 5 | 200* | 18.0% | 82.0% | 61.5% | 37.5% |
| 6 | 1,000 | 46.9% | 53.1% | 85.3% | 13.8% |
| 7 | 2,000 | 56.3% | 42.9% | 91.3% | 5.8% |
| 8 | 2,000 | 75.8% | 23.4% | 93.0% | 3.6% |
| 10 | 2,000 | 87.0% | 10.6% | 95.7% | 1.7% |
| 12 | 2,000 | 88.9% | 7.4% | 96.5% | 1.1% |
* Small samples. Positions with few pieces placed take far longer to solve exactly (with 4 pieces placed, about 12 seconds each), so read those rows as a direction, not a precise rate. The remaining share in each row is a forced loss for the player to move. Results for 9 and 11 pieces placed are in the raw data.
Three things stand out:
- Classic endgames are drawish much longer. With six pieces placed, just over half of the random positions (53.1%) are still draws under classic rules. With squares counting, only 13.8% are.
- With squares, positions get sharp early. Even with four or five pieces on the board, squares turned many classic draws into forced wins. Only 24 positions were solved at four pieces, so that row is indicative only.
- Squares don't only help the player to move. Across all buckets, forced losses rose from 264 positions under classic rules to 342 with squares. More winning shapes means more ways to be squeezed, too.
Finding 2: safe pieces run out much faster with squares
A piece is safe to hand over if your opponent can't win with it anywhere on the board. We counted safe pieces in 20,000 random positions at each stage:
| Pieces placed | Classic: pieces safe | Squares: pieces safe | Classic: no safe piece | Squares: no safe piece |
|---|---|---|---|---|
| 4 | 92.4% | 86.6% | 0% | 0% |
| 6 | 70.6% | 56.5% | 0.5% | 2.5% |
| 7 | 56.9% | 40.2% | 3.0% | 11.0% |
| 8 | 43.8% | 26.7% | 10.0% | 27.5% |
| 10 | 24.5% | 10.4% | 37.1% | 66.1% |
| 12 | 16.6% | 5.3% | 59.8% | 85.0% |
With eight pieces placed and squares counting, more than a quarter of random positions leave no safe piece at all, and the player who has to hand one over has already lost. Under classic rules the same happens in one position in ten. That's why keeping a safe piece in reserve is the core skill in our strategy guide.
Finding 3: how the Quads AI does against itself
We also had the computer opponents that ship in Quads play each other: 5,000 games per setting, 20,000 in total. The AI makes the same move every time in a given position, so we randomised the opening turns to get varied games. Every game uses 2×2 squares. "First placer" is the player who places the first piece, which means they received the very first hand-off.
| Level | Random opening turns | First placer wins | Other player wins | Draws |
|---|---|---|---|---|
| Grandmaster vs Grandmaster | 2 | 53.0% | 19.4% | 27.6% |
| Grandmaster vs Grandmaster | 4 | 48.5% | 29.9% | 21.6% |
| Skilled vs Skilled | 2 | 40.8% | 40.3% | 18.9% |
| Skilled vs Skilled | 4 | 42.9% | 42.9% | 14.1% |
Games averaged about 12.5 placements, so most end with three or four pieces unplayed. Grandmaster's games draw more often than Skilled's, and they lean clearly toward the player who places first. Skilled's games are almost exactly even. We haven't worked out why yet, so treat it as an observation about our current AI, not a fact about perfect play. The AI only searches exactly in the last part of the game and uses rules of thumb before that. Also, with just two random opening turns some openings repeat, so that row is less varied than 5,000 games suggests.
What this means when you play
- If you want decisive games, play with squares. The classic rules leave a lot more room to hold a draw. With squares, the middle of the game is where it's won and lost.
- The danger zone is 6 to 8 pieces placed. That's where safe pieces go from plentiful to scarce, especially with squares. Slow down and count.
- A perfect draw needs perfect play. The solved result says a draw is always available. It doesn't say you'll find it. Practise spotting the squeeze in the daily puzzle.
Frequently asked questions
Is Quarto solved?
Yes. Luc Goossens solved classic Quarto by computer in 1998: with perfect play from both players, the game is a draw. Neither player can force a win from the empty board.
Does the first player have an advantage in Quarto?
Not with perfect play, because the solved result is a draw. In practice, games between imperfect players are usually decided by who is first forced to hand over a winning piece.
Does the 2×2 square variant change the result?
Our study didn't re-solve the full game with squares, but squares change endgames a lot. In 2,000 random positions with seven pieces placed, 42.9% were draws under classic rules and only 5.8% with 2×2 squares counting.